park-and-ride: good for the city, good for the region?

Regional Science and Urban Economics 41 (2011) 455–464

Contents lists available at ScienceDirect

Regional Science and Urban Economics

j ourna l homepage: www.e lsev ie r.com/ locate / regec

Park-and-ride: Good for the city, good for the region?

Vladimir Karamychev, Peran van Reeven ⁎Erasmus University Rotterdam and Tinbergen Institute, The Netherlands

⁎ Corresponding author at: Erasmus School of EcRotterdam, Burg. Oudlaan 50, NL-3062 PA Rotterdam,408 2113; fax: +31 10 408 9153.

E-mail address: [email protected] (P. van Reeve1 This type of P+R is in contrast to “remote” and “su

drivers close to their place of origination.

0166-0462/$ – see front matter © 2011 Elsevier B.V. Aldoi:10.1016/j.regsciurbeco.2011.03.002

a b s t r a c t
a r t i c l e i n f o
Article history:Received 18 August 2010Received in revised form 4 March 2011Accepted 7 March 2011Available online 12 March 2011

JEL classification:R41R48

Keywords:Park and rideTrafficTransportation modesDiscrete choice

At the edge of cities, park-and-ride (P+R) facilities pop up with the aim to intercept motorists from travelinginto the city. However, these facilities also appear attractive to public transport users who start using theircars for getting to the P+R location. This paper analyzes the overall impact of P+R on total car traffic andsocial welfare by means of a discrete modal choice model. The results show that the distribution ofindividuals' preferences for car over public transport is the main determinant of this impact. P+R has a largertraffic reducing effect if more individuals prefer their car. At the same time, the shift of traffic from city toperiphery improves welfare. These effects get stronger when a P+R facility provides a superior access to themainline public transportation network.

onomics, Erasmus UniversityThe Netherlands. Tel.: +31 10

n).burban” P+R that aim to stop

l rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

On the edge of cities and towns, more and more parking facilitiespop upwith direct access to a public transport service. These so-calledpark-and-ride (P+R) facilities intercept motorists from traveling intothe city, close to their final destination, and are popular throughoutthe United States and Europe. This is the most common type of P+Rand sometimes referred to as “peripheral” or “local” P+R (AASHTO,2004).1

The popularity of P+R among cities is not without a reason. First,it improves accessibility. Most cities suffer from congestion; they areoften physically constrained to increase the capacity of the roadnetwork and the parking stock in the city center. P+R increases thenumber of parking places while avoiding the construction of new carparks in the urban core.

Second, by encouraging people to take public transport for part oftheir trips, P+R facilities help to alleviate traffic congestion and otheradverse external effects of travel by private car. Any reduction incongestion from the transfer of motorists to P+R frees road space inthe city and may induce further visitors that stimulate economicactivity.

Third, opening P+R facilities along existing public transportnetworks increases public transport ridership and may improve thecost recovery of those services. For example, current fare revenues ofurban public transport fall short of the operational costs by 66% inNorth America (Federal Transit Administration, 2008) and by anaverage of 48% in Europe (UITP, 2005). Moreover, increased publictransport ridership allows for further improvements in quality ofservice.

Finally, as an urban transport policy, P+R is also generally saleableto the public. It widens the choice of transport options, not forcingpeople out of their cars when using a car is their preferred option.P+R facilities integrate the private car into the public transportsystem, allowing motorists to evade the low speeds of inner citydriving, the inevitable congestion delays, and the high costs of parkingin the city, while enjoying the convenience and comfort of theirprivate car for the larger part of the journey outside the city.

Despite its popularity, P+R has drawn little scientific interest. Asmall body of literature analyzes the planning and the design of P+Rfacilities. Wang et al. (2004) and Horner and Groves (2007) analyzeP+R locations with different objective functions. Wang et al. (2004)consider P+R as an investment decision and focus on profitmaximization and social cost minimization. Horner and Groves(2007) take a traffic engineering approach and capture the max-imum number of vehicles early in their journeys. Given a P+Rlocation, García and Marín (2002) analyze capacity and pricingdecisions that minimize total travel costs in a mathematical pro-gramming approach. Bos and Molin (2006) carry out an experimentwith incentives that may increase P+R usage.

http://dx.doi.org/10.1016/j.regsciurbeco.2011.03.002

mailto:[email protected]

http://dx.doi.org/10.1016/j.regsciurbeco.2011.03.002

http://www.sciencedirect.com/science/journal/01660462

r1 R

Fig. 1. The city (inner circle) and the periphery (shaded ring).

456 V. Karamychev, P. van Reeven / Regional Science and Urban Economics 41 (2011) 455–464

The motivation for this paper comes from another body ofliterature on P+R. This literature is more descriptive and policyoriented, and focuses on the objectives behind P+R and the actualrealization thereof. Based on a review of policy documents and impactstudies (Cairns, 1997; Parkhurst, 2000; Meek et al., 2008a, 2008b),and on the results drawn from surveys and interviews (Parkhurst,1995 and Meek et al., 2009, 2010), this literature calls into questionthe role of P+R in reducing car traffic. The central argument is thatthe incentives offered to motorists also apply to existing publictransport users. By shifting modes, they may enjoy the benefits ofmotoring for the P+R access trip.

The degree to which this negates mileage savings made frominterceptedmotorists is likely to be considerable, especially if one takesinto account that access journeys are generally longer than the trip legbetween the P+R location and theurban center (Parkhurst, 2000,Meeket al., 2008b). Empirical evidences show that a significant proportion ofP+R users may indeed come from public transport (see Meek et al.,2008b). This has brought the role of P+R in reducing car usage intoquestion. The American Association of State Highway and TransportOfficials is critical aboutP+Rnear theplace of destinationandprefers tolocate it close to the origin of trips (American Association of StateHighway and Transportation Officials, AASHTO, 2004). This preferenceis also explicit in the approach of Horner and Groves (2007) and therecommendations of Parkhurst and Richardson (2002).

In this paper, we consider themost common type of P+R, which islocated at the edge of the city, and analyze its impact on total cartraffic and social welfare. In a discrete choice model, we analyze theeffect of opening P+R on themodal choice of individuals who alreadytravel into the city without P+R. Individuals are assumed heteroge-neous in their geographical location as well as in the costs of using theprivate car and public transport. They choose one of the followingthree modes of transportation: the car, public transport, or thecombination of the two with a transfer at a P+R facility. The latter islocated along the existing public transport network at the edge of thecity. The model accounts for the generalized transportation costs foreach of these three modes. Differences in these costs characterizeindividuals' preferences for one mode over another.

In the next step, we extend this basic model to accommodate fornegative congestion externalities. Individuals experience instanta-neous disutility from traffic at any point on their route. This convertsthe discrete choice model into a rational expectations model, whereindividuals' initial belief about traffic intensities along the routeinduces such a modal split that the actual realized traffic intensitiesequal to the expected ones.

Our results are as follows. The model identifies two reasons whyindividuals use a P+R facility. First, P+R may be cheaper than usingthe private car in the city center. This is the case when the city has allthe usual problems associated with car use, such as congestion andparking problems. Second, P+R may provide a cheaper access to themainline public transport network than the place of origination. Thisis the case when individuals use P+R to avoid relatively slow and lowfrequency local services at their home locations.

These reasons result in opposite patterns of modal split. If P+Rcosts less than using the car in the city center, then P+R attracts thoseindividuals who prefer the car to public transport and who reside farfrom the P+R location. These individuals benefit from avoiding thecongestion and parking costs in the city center, while enjoying theirprivate car for the larger part of their journey. If, in contrast, P+Rprovides cheaper access to the mainline public transport network, itattracts those individuals who prefer public transport to the car andwho reside close to the P+R location. These individuals enjoy P+R asa more efficient public transport access point, while not incurring toomuch disutility from having to drive their car to the P+R facility.

When both reasons apply simultaneously, P+R attracts indivi-duals from all over the periphery, which yields the largest reduction incar traffic, and can be achieved by making P+R as cheap and efficient

as it is ever possible. This is in contrast to the suggestion of Parkhurst(1995) and Meek et al. (2008a) to make P+R more expensive thanpublic transport in order to prevent individuals from switching frompublic transport to their private car.

In the presence of congestion, P+R has yet another effect onmodal split. By shifting traffic away from the city into the periphery, itreduces congestion, and makes the private car more attractive thanpublic transport for individuals who reside next to the city border. Asa result, some individuals switch directly from public transport to thecar for their trip into the city center. This effect vanishes if the costof using P+R is low. Hence, this effect constitutes another reasonwhyP+R should be made as cheap and as efficient as possible.

P+R has a favorable effect on welfare beyond the reduction intraffic. The reason is that P+R expands individuals' choices, and,consequently, increases users' consumer surplus. Besides, a shift oftraffic into the periphery is likely to have an additional positivewelfare effect since in most cities the external costs of driving in thecity are higher than in the periphery.

The positive effects of P+R on traffic andwelfare hold under fairlygeneral assumptions, e.g., when the distribution of preferences isunimodal and the private car dominates the modal split. This latterassumption is empirically validated in most cities; see Urban Audit2004 (Eurostat, 2008). However, an empirical study into the shape ofthe distribution function and the actual modal split remains useful inassessing whether a new P+R facility will indeed reduce car traffic.

The rest of the paper is organized as follows. Section 3 specifies themodel, which is analyzed in Section 4. Section 5 concludes.

2. Model

We consider a city and its circular periphery (region); see Fig. 1.This city is in the form of a disk and has unit radius. The periphery hasthe form of a ring around the city and has an outer radius RN1. A unitmeasure of individuals lives in the periphery. Every individual i ischaracterized by its location at distance ri∈ [1,R] from the city. Thespatial distribution density function of individuals has support [1,R]and is denoted by fr(r).

Each individual is a traveler who wants to reach the city center byusing one of the following three transportation modes. The first modeis public transport, denoted by superscript (P). An individual i, whotravels distance ri to the city center by public transportation, gets thefollowing utility

U Pð Þ = U0;i−riti− a; ð1Þ

where U0, i is his personal gross utility of going to the city center (forwork, leisure, or any other reasons), and tiN0 and ãN0 are the variableand fixed costs correspondingly. The variable cost ti covers thedistance related part of the tariff and an individual's valuation of in-vehicle time. The fixed cost ã is the fixed part of the tariff, the timecosts associated with getting to a network access point, the waitingtime for departure, etc.

457V. Karamychev, P. van Reeven / Regional Science and Urban Economics 41 (2011) 455–464

The second mode is private car, denoted by superscript (C). Ifindividual i travels the distance ri to the city by car, he gets utility

U Cð Þ = U0;i−ripi−c:

Here, piN0 and c N 0 are variable and fixed costs correspondingly. Thevariable cost pi includes the cost of the individual of driving a privatecar per unit of distance, as well as his valuation of the travel timeinvolved. The cost c is the additional fixed cost of using the car in thecity, which covers such items as lower average speeds of driving a carin the city, congestion delays, searching for parking space, parkingfees, etc.

The third mode is a combination of private and public transpor-tation with a transfer at a P+R facility. In this mode, denoted bysuperscript (P+R), individual i travels the distance (ri−1) from hisresidence to the city border by car, switches to public transportand travels the remaining unit distance to the city center by publictransport. Therefore, the utility of individual iwho uses P+R to travelthe distance ri into the city center is given by

U P + Rð Þ = U0;i− ri−1ð Þpi−s−ti;

where s N 0 represents a fixed cost of making a transfer from car topublic transport at a P+R facility. This cost covers the transfer time,the discomfort of walking to the point of service, the waiting time forpublic transport to arrive, possibly a P+R parking fee, etc.

For the sake of convenience, we define the following net utility andcosts:

ui ≡ ti−pið Þ; s≡ s−að Þ; and c≡ c−að Þ:

Utility ui is the net utility of individual i of traveling a unit distance bycar rather than by public transport. Alternatively, ui can be interpretedas the opportunity cost of individual i of traveling a unit distance bypublic transport. Some individuals prefer the car to public transport,whereas for others it is the other way around. We assume that netutility ui of individual iwith location ri=r is distributed in accordancewith the (conditional on r) distribution density function fu(u|r) overthe support u;u½ �, where u b 0 b u. The conditional distribution func-tion is denoted by Fu(u|r). The shaded area in Fig. 2 represents the setof individuals in the (r,u) space.

Net cost of using P+R s is the opportunity cost of access to thepublic transport network at a P+R location rather than at the place oforigination. Finally, net city driving cost c is the opportunity cost ofentering the city by car rather than by public transport. In principle,both c and s can be positive or negative.

In our model, each individual has his own net utility ui and weassume that the variation of ui is sufficiently wide so that in absolutevalues, u and ū are larger than both c and s. This is a simplifying

u

0r

u

R

u

1

Fig. 2. Set of individuals (vertically shaded area) in (r,u) space.

assumption that reduces the number of different cases that we have toconsider in themodel. For the same sake of simplicity, we assume thatboth P+R cost s and the city driving cost c are identical for allindividuals.

Finally, we assume that the personal gross utility of traveling intothe city U0, i is sufficiently large. In other words, all individualsnecessarily travel into the city center by choosing one of the threemodes. This is a simplifying assumption that allows us to focus on theeffect of opening P+R on individuals who already travel by car orpublic transport. The resulting choice model involves only threealternatives and is incomparably simpler than choosing from fouralternatives, where the fourth option is not to travel.

Every individual chooses the transportation mode that maximizeshis utility. This choice is not affected by the personal gross utility oftraveling into the city because U0, i is in the utility specification for allthree alternative modes. Hence, we are free to set

U0;i = riti + a; ð2Þ

so that the utilities of an individual (r,u) can be written in terms of netutility and costs as follows:

U Pð Þ = 0; U Cð Þ r;uð Þ = ru−c; and U P + Rð Þ r;uð Þ = r−1ð Þu−s:

The choices of all individuals generate demands for private car, publictransport and the combination thereof through the transfer at a P+Rfacility. We denote the sets of individuals who travel by car and P+Rmode by S(C) and S(P+R):

S Cð Þ = r;uð Þ��U Cð Þ r;uð Þ≥max U Pð Þ

;U P + Rð Þ r;uð Þ� �n o

; and

S P + Rð Þ = r;uð Þ��U P + Rð Þ r;uð Þ≥max U Pð Þ

;U Cð Þ r;uð Þ� �n o

:

The demands for these modes are denoted by D(C), D(P), and D(P+R):

D Cð Þ = Pr ri;uið Þ∈S Cð Þ� �; D P + Rð Þ = Pr ri;uið Þ∈S P + Rð Þ� �

; and

D Pð Þ = 1−D Cð Þ−D P + Rð Þ:

The expected distances that the car and P+R users travel using thecar are denoted by M(C) and M(P+R):

M Cð Þ = D Cð Þ⋅E r�� r;uð Þ∈S Cð Þ� �

and

M P + Rð Þ = D P + Rð Þ⋅E r−1�� r;uð Þ∈S P + Rð Þ� �

:

The total car mobility M is defined as the aggregate distance traveledby car by all individuals:

M = M Cð Þ + M P + Rð Þ:

We also make a distinction between inner city M(IN) and peripheraltraffic M(OUT). Because the inner city traffic comes from car users thatdrive unitary distance, and because all the remaining traffic isperipheral, we define M(IN) and M(OUT) as follows:

M INð Þ = 1⋅D Cð Þ = D Cð Þ and M OUTð Þ = M−M INð Þ = M P + Rð Þ + M Cð Þ−D Cð Þ:

Individuals' aggregate net consumer surplus CS is defined by

CS ≡ E max U Pð Þ;U Cð Þ r;uð Þ;U P + Rð Þ r;uð Þ

� �� ;

where the expectation is taken over the set of all individuals. Finally,the social welfare SW generated in the whole economy consists ofthe consumer surplus CS and the externalities H(C)(M(C)) and H(P+R)

2 For finite values of R, this condition is s−cb− c−a ð Þ = Rb0.


(M(P+R)) that the private traffic imposes on the city and the peripherycorrespondingly:

SW = CS + H Cð Þ M Cð Þ� �+ H P + Rð Þ M P + Rð Þ� �

: ð3Þ

Our objectives are as follows. First, we derive the parameters of themodel underwhich individuals use P+R.We establish two reasonswhyP+R can be attractive. Depending onwhich reason applies, andwhetherboth apply simultaneously, we derive three possible patterns of theresulting modal split. Second, for a given pattern of the modal split, weanalyzehow the introductionof the P+R facility changes themodal split,and, more importantly, the total car traffic M, consumer surplus CS, andsocial welfare SW. This analysis accounts for the externalities of car usageon the society in general, such as noise and pollution. Besides, we discusstheoptimal locationof P+R. Third,weaccommodate for theexternalitiesthat the road users exhibit on themselves, that is, congestion, and showhowa congestionexternality changes the aforementionedeffects of P+Ron the modal split, car traffic, and social welfare. Finally, we discuss howendogenous demand changes the results.

3. Analysis

The analysis of the model is structured as follows. Subsection 3.1analyzes the demand for P+R and identifies two main reasons forP+R usage. Subsections 3.2 and 3.3 analyze the impact of P+R onmodal split and total car traffic respectively. Subsequent subsectionsextend and generalize the basic analysis in different directions.

We begin the analysis by identifying individuals in the space (r,u)whoare indifferentbetween someof themodes.Wedefine(i)u(P=C)(r),(ii) u(P+R=C)(r), and (iii) u(P=P+R)(r) as the net utilities of individualsresiding atdistance r from the citywhoare indifferentbetween (i) publictransport and car, (ii) P+R and car, and (iii) public transport and P+Rrespectively. Formally,

U Cð Þ r;u P=Cð Þ rð Þ� �

≡U Pð Þ;

U Cð Þ r;u P + R=Cð Þ rð Þ� �

≡U P + Rð Þ r;u P + R=Cð Þ rð Þ� �

;and

U Pð Þ≡U P + Rð Þ r;u P=P + Rð Þ rð Þ� �

;

so that the indifferent travelers are given by

u P=Cð Þ rð Þ = cr; u P + R=Cð Þ rð Þ = c−s; and u P=P + Rð Þ rð Þ = s

r−1:

We also define r* such that the individual (u(P+R=C)(r*), r*) isindifferent between all three modes. It follows that

r� =c

c−s;

so that u(P=C)(r*)=u(P+R=C)(r*)=u(P= P+R)(r*)=c− s by con-struction. The existence of such r* highlights the difference betweenthis model and a model with four alternatives where individuals havethe option not to travel. In the latter case, there are 6 indifferencecurves and 15 generic pairwise intersection points. This significantlycomplicates the model and obscures the results.

3.1. Two reasons for using a P+R facility

Depending on the model parameters, P+R may be unattractive toindividuals. In the following proposition we identify parameterrestrictions under which P+R is used so that D(P+R)N0.

Proposition 1. A P+R facility is used, i.e., D(P+R)N0, only in thefollowing two cases:

(i) 0bsb(1−1/R)c, and(ii) sb0.

Proof. Suppose that an individual (r,u) prefers P+R to car andpublic transport, i.e., U(P+R)(r,u)NU(C)(r,u) and, at the same time,U(P+ R)(r,u)NU(P)(r,u):

r−1ð Þu−s N ru−cr−1ð Þu−s N 0 :

�

This system can be rewritten as follows:

u∈ sr−1ð Þ ; c−s

� �= u P=P + Rð Þ rð Þ; u P + R=Cð Þ rð Þ� �

:

This can only be satisfied when c−sNs/(r−1) (we disregard non-generic equality cases).Suppose that sN0, then assuming (1−1/R)c≤s and cN0 yields:

sr−1ð Þ ≥ s

R−1ð Þ =sR

R−1ð Þ−s =s

1−1 = Rð Þc ⋅c−s≥ c−s;

so that no such (r,u) exists. Hence, it must be that either c≤0, whichimmediately yields the contradictory inequality c− sb0bs/(r−1), or0bsb(1−1/R)c which is case (i).

Suppose, to the contrary, that sb0. Then, for any value of c, thecondition u∈(s/(r−1), c−s) can be satisfied for r close enough toone, which is case (ii). □

Proposition 1 can be interpreted quite easily when we assumethat the size of the periphery is much larger than city radius, so that1/R≈0. In this case, there are two reasons why individuals may wantto use P+R. The first reason is that P+R is cheaper than using the carin the city, i.e., when s−c b 0.2 The second reason is that P+R mayoffer a more efficient access to the mainline public transport networkthan the place of origination, i.e., when s−a b 0. When, to thecontrary, s−c N 0 and s−a N 0, i.e., when P+R is attractive neither asa parking place nor as a public transport access point, individualsdo not use P+R and choose either the car or public transport.

3.2. Modal split

The motivation for building P+R facilities is to alleviate theburden of congestion and the wider impact of car usage on the urbanenvironment. In order to intercept motorist, P+R is often offered freeof charge (or at very low costs) at public transport access locationswith an efficient and high-frequency service into the center, such as(light-)rail and shuttle buses. In this case, P+R is cheaper than usingthe car in the city, i.e., when s−ab0 and, at the same time, providescheaper access to the mainline public transport network than theplace of origination, i.e., when s−cb0, so that sbmin(0,c). Fig. 3illustrates the resulting modal split.

The bold curve u(P=C)(r) in Fig. 3 represents individuals who areindifferent between using the car and public transport for the wholetrip; the car is more attractive for all individuals above u(P=C)(r), andpublic transport is more attractive for all individuals below this curve.Similarly, the dashed horizontal line u(P+R=C)(r) represents indivi-duals who are indifferent between P+R and the car, and the dashedcurve u(P=P+R)(r) represents individuals who are indifferent be-tween public transport and P+R.

As a result, the shadedarea represents individualswhodrive their carsto the P+R facility and transfer to public transport for the trip into thecity. In particular, the vertically shaded area represents individualsattracted by P+R from the car mode, and the horizontally shaded arearepresents individuals attracted from public transport. Hence, P+Rattracts individuals from all over the region, nearby as well as distant,irrespective of the sign of ui. Fig. 3 also shows that no individual in the

Fig. 3. Individuals' choices for sbmin(0,c).

Fig. 5. Individuals' choices for cb sb0.


vicinity of the city border uses public transport. For such individuals, P+Ris the superior access point to the public transport network.

In the rest of the analysis, we assume that r ≡ 1 + s= uÞbRð so thatat least some individuals use public transport for the whole trip. Fornotational convenience, we define r*=R so that r* is the location ofthe farthest P+R users when sbmin(0,c).

An argument against providing free or very cheap P+R is that thisalso attracts public transport users. Using this argument, Parkhurst(1995) and Meek et al. (2008a) suggest making P+R less attractivethan public transport so that the resulting price hierarchy preventsindividuals from shifting from public transport to the private car forthe P+R access journey. In our model, this occurs when, first, P+Rcost is lower than the cost of using the car in the city, i.e., when s−cb0,and, second, P+R provides more expensive access to the publictransport network than the place of origination, i.e., when s−a N 0, sothat cNsN0. Fig. 4 illustrates the modal split in this case.

Fig. 4 shows that expensive P+R prevents individuals with apreference for public transport, i.e., with uib0, to use P+R. Theyrather access public transport at the place of origination. Only in-dividuals with a preference for car, i.e., with uiN0, who reside at asufficient distance from the city, may find P+R attractive. The reasonis that the P+R access journey has to be sufficiently long, so thatbenefits of driving the car compensate for the higher costs of publictransport access at a P+R location. Indeed,u(P=P+R)(r)bu(P+R=C)(r)only for rNr*=c/(c−s).

Some of these individuals with a preference for the private carcome from public transport, as shown by the horizontally shaded area.These individuals used public transport due to the high costs of citydriving, but the opening of a P+R facility gives them incentives toreturn to their most preferred mode, i.e., the car, for the P+R accessjourney. Hence, the price hierarchy cNsN0 does not prevent publictransport users to shift to the private car.

The case where cNsN0 has its mirror image in the case where P+Rprovides cheaper access to the mainline public transport networkthan the place of origination, i.e., when s−ab0, but is more expensivethan using the car into the city, i.e., when s−c N 0, so that cbsb0. Thiscase describes an efficient P+R facility in an atypical city that hardlyhas any of the familiar problems associated with car usage and ample(free) parking. Fig. 5 illustrates the modal split in this case.

Fig. 5 shows that P+R has no appeal to thosewho prefer car usage,i.e., with uiN0, because driving in the city is cheaper than using P+R.P+R only attracts individuals with a preference for public transport,i.e., with uib0, insofar these individuals reside sufficiently close to theP+R facility. The reason is that the P+R access journey has to beshort so that cheap P+R fully compensates the disutility of having todrive the car to the P+R location. For notational convenience, wedefine r* in case cbsb0 as follows

r� = min−cs−c

;Rn o

;

so that r* is the location of the farthest P+R users.

3.3. The impact of P+R on total car traffic

Opening the P+R facility has two effects on existing car traffic.First, the motorists denoted by the vertically shaded areas in Figs. 3–5transfer to public transport at the P+R location. By doing so, eachmotorist drives his car a unit of distance less, which reduces car trafficin the city. Second, the public transport users, denoted by thehorizontally shaded areas in Figs. 3–5, shift to the car for the trip fromthe place of origination to the P+R location. By doing so, an individual(ri,ui) starts driving the distance (ri−1) by car, which increases totalcar traffic outside the city. The balance between these two effectsdetermines whether P+R increases or decreases total car traffic M.

In the casewhere the car is more costly than public transport in thecity center, i.e., when cN0, there will be no P+R usage if s≥(1−1/R)c. Similarly, in the cases where the car is cheaper, i.e., when cb0, therewill be no P+R usage if P+R provides more costly access to thepublic transport network than the place of origination, i.e., s≥0. Thisallows us to define

s ≡ max 0; 1−1R

� �c

� �;

and model the absence of P+R by formally setting s = s.We analyze M as a function s by computing its derivative dM/ds.

When dM/dsN0, opening a P+R facility, which is equivalent to areduction of s, reduces total car trafficM. On the other hand, when dM/dsb0, opening the P+R increases total car traffic M(s).

Since the modal split in the case where P+R provides cheap andefficient access to the mainline public transport network, i.e., wheresb0, has a different pattern from the case where it provides morecostly access than the place of origination, i.e., where sN0, we treatthem differently.

Fig. 4. Individuals' choices for 0b sb(1−1/R)c.


Proposition 2. For 0bsb(1−1/R)c:

dMds

= D P + Rð Þ⋅E ddu

lnfu u jrð Þ j r; uð Þ∈S P + Rð Þ� �

:

Proof. Computing M directly yields:

M = ∫r⁎

1

r ∫u

u P = Cð Þ rð Þfu u jrð Þdu

0@

1Afr rð Þdr

+ ∫R

r�r−1ð Þ ∫

u P + R= Cð Þ

u P= P + Rð Þ rð Þfu u jrð Þdu + r ∫

u

u P + R= Cð Þfu u jrð Þdu

0@

1Afr rð Þdr:

Differentiating M with respect to s yields

dMds

= ∫R

r�

dds

r−1ð Þ ∫u P + R= Cð Þ


u


0@

1Afr dr

= ∫R

r�r−1ð Þ du P + R=Cð Þ

dsfu u P + R=Cð Þ jr� �

−du P=P + Rð Þ

dsfu u P=P + Rð Þ jr� � !

fr dr−

−∫R

r�rdu P + R=Cð Þ

dsfu u C=P + Rð Þ jr� �

fr dr

= ∫R

r�fu u P + R=Cð Þ jr� �

−fu u P=P + Rð Þ jr� ��

fr rð Þdr

= ∫r; uð Þ∈S P + Rð Þ

ddu

lnfu u jrð Þ� �

fu u jrð Þfr rð Þdudr

and the statement of the proposition follows. □

In accordance with Proposition 2, the two effects of s on M are ofthe same magnitude, and cancel each other when the distribution ofutilities is uniform, i.e., fu u jrð Þ = 1= u−uð Þ. In this case, dM/ds=0,and opening a P+R facility does not affect total car traffic. If, however,fu(u|r) monotonically increases or decreases, so doesM(s). As a result,opening P+R facility decreases car traffic if the utility distributionfunction increases, and increases car traffic if this function decreases.For non-monotonic distributions, the slope ofM is the average slope ofln fu(u|r) over the set of individuals that use P+R.

The reason why opening a P+R facility has no impact on total cartraffic for a uniform distribution of individuals' utility is as follows.Motorists who are attracted by themarginal decrease in s are all locatedalong the indifference curve u(P+R=C). The number of such indifferentmotorists located at any distance r is determined by the derivative of(−u(P+R=C)) with respect to s, which is −∂u(P+R=C)/∂s=1. Eachsuchmotorist drives one unit of distance less, so that in aggregate, theydecrease total car traffic by−1⋅∂u(P+R=C)/∂s=1, i.e., by one.

On the other hand, public transport users who are attracted by themarginal decrease of s are all located along the indifference curveu(P=P+R). The number of such indifferent motorists located at thesame distance r is determined by the derivative of u(P=P+R) withrespect to s, which is ∂u(P=P+R)/∂s=1/(r−1). Each such individualstarts driving the distance (r−1) from his origin to P+R location, sothat in aggregate, they increase total car traffic by (r−1) ⋅∂u(P=P+R)/∂s=1, i.e., also by one. Therefore, the extra traffic generated by thepublic transport users residing at r is equal to the traffic reduction bythe motorists also residing at r. As this happens for all r∈[r*, R], theoverall effect is absent.

When the distribution of individuals' utilities has an increasingdensity, the number of individuals along the indifference curveu(P+R=C) is relatively (with respect to the uniform distribution)higher than along the indifference curve u(P=P+R). That is why thetraffic reduction from the former is larger than the extra trafficgenerated by the latter, so that dM/dsN0.

Contrary to the case sN0, the effect of P+R cost s on total carmobility M(s) has a positive, i.e., car traffic reducing, bias when sb0.The following proposition demonstrates this.

Proposition 3. For sb0:

dMds

= ∫r

1

fu u C=P + Rð Þ jr� �

fr rð Þdr

+ ∫r; uð Þ∈Sr≥r

ddu

lnfu u jrð Þ� �

fu u jrð Þfr rð Þdudr:

Proof. Computing M directly yields:

M Cð Þ = ∫r

1

r−1ð Þ ∫u P + R= Cð Þ

ufu u jrð Þdu + r ∫

u


0@

1Afr rð Þdr +

+ ∫r

r�r−1ð Þ ∫

u P + R= Cð Þ


u


0@

1Afr rð Þdr

+ ∫R

r�r ∫

u

u P = Cð Þfu u jrð Þdu

0@

1Afr rð Þdr

Differentiating M with respect to s yields

dM Cð Þ

ds= ∫

r

1

fu u P + R=Cð Þ��r� �fr rð Þdr

+ ∫r

r�

fu u P + R=Cð Þ��r� �−fu u P=P + Rð Þ rð Þ

��r� �� fr rð Þdr;

and the statement of the proposition follows. □

The expression for the slope ofM(s) consists of two parts. The firstpart is always strictly positive. This positive effect stems from thesimple fact that P+R provides superior access to the public transportnetwork for individuals with ri∈ 1;r ð Þ. The second part vanishes for theuniform distribution of individuals' net utilities fu(u|r), and mono-tonically increases or decreases if the utility distribution function doesso.

Proposition 2 and 3 together explain that ceteris paribus, thedecrease in s generates a greater reduction of total car traffic when aP+R facility provides a superior access to the mainline publictransportation network. In particular, when sb0, P+R can be atraffic-reducing measure when fu(u|r) is a (slightly) decreasingfunction, which never happens when sN0. In other words, loweringthe cost of using P+R (parameter s) makes P+R more attractive atan increasing rate, so that we can view P+R as an increasing returnto scale (IRS) traffic-reducing technology.

3.4. Optimal location of P+R

The relative strength of the two reasons for using a P+R facility isthe main determinant of the optimal location of P+R. Suppose, first,that the only reason for using P+R is to save on the costs of drivinginto the city center, i.e., cNsN0 and both c and a are large. In this case,r* is close to R and only few individuals use P+R. Then, the impact ofP+R on traffic, whether it is positive or negative, is maximal whenthe P+R facility is located at the city border. Moving P+R furtherinto the periphery eliminates all P+R usage, and, therefore nullifiesits effect on traffic; see Fig. 4.

In the opposite case, when the only reason for using P+R is to saveon the costs of public transport access, i.e., cbsb0. In this case, r is closeto R, and only few individuals use public transport. The number of

3 The most general externality specification can be obtained by using a generalkernel instead of the Dirac delta-function. The analysis that follows remains valid forany kernel.


individuals that use public transport for the whole journey into thecity center can be maximized by locating P+R at the city border, seeFig. 5.

Hence, the validity of the argument of the American Association ofState Highway and Transport Officials, Horner and Groves (2007) andParkhurst and Richardson (2002) to locate P+R close to the originof trips, i.e., deep in the periphery, requires a negative overall effect ofP+R on traffic and is not correct otherwise.

3.5. The impact of P+R on social welfare

Opening P+R has a positive effect on individuals' aggregateconsumer surplus CS. All P+R users increase their utility, and theother individuals remain unaffected by P+R. We analyze CS as afunction of s by computing its derivative dCS/ds. The followingproposition states the result.

Proposition 4.dCSds

= −D P + Rð Þ

Proof. This is a fundamental result in the utility maximization theory,which we also show explicitly:

CS = ∫r�

1

∫u P= Cð Þ rð Þ

u0⋅fu u jrð Þdu + ∫

u

u P = Cð Þ rð Þru−cð Þfu u jrð Þdu

0@

1Afr rð Þdr +

+ ∫R

r�

∫

u P= P + Rð Þ rð Þ

u0⋅fu u jrð Þdu + ∫

u P + R= Cð Þ

u P= P + Rð Þ rð Þr−1ð Þu−sð Þfu u jrð Þdu

+ ∫u

u P + R= Cð Þru−cð Þfu u jrð Þdu

!fr rð Þdr;

and

dCSds

= ∫R

r�∫

u P + R= Cð Þ

u P = P + Rð Þ rð Þ

dds

r−1ð Þu−sð Þfu u jrð Þdu0@

1Afr rð Þdr = −D P + Rð Þ

:

□

Because opening P+R decreases traffic inside the city M(IN) andincreases traffic in the peripheryM(OUT), it generates opposite welfareeffects in the city and in the periphery. In general, car traffic hasseveral positive and negative effects on the rest of the economy. Theexternal social benefits of traffic are increased economic activity,employment, etc., whereas the costs are increased pollution, noise, etc.Most often, the introduction of P+R ismotivated by the latter, andweassume that traffic has a negative net effect on SW, i.e., dH(C)/dM(C)b0and dH(P+R)/dM(P+R)b0 in (3). Next, as the external costs of carusage are generally higher in the city than in the periphery (UNITE,2003), we assume that dH(C)/dM(C)bdH(P+R)/dM(P+R)b0.

In this case, any monotonically increasing distribution of indivi-duals' utilities leads to an increase in welfare in response to theopening of a P+R facility. This follows from the impact of P+R on Mas is previously shown, and the positive impact of P+R on consumersurplus CS.

3.6. Congestion

So far, the utility of car travelers has been independent of thetraffic levels in the city and the periphery. In this subsection, we adjustthe model in order to take into account congestion externalities. Forsimplicity, we assume that only private (car) traffic generates thisexternality, and individuals that use the public transport networkneither impose externalities on themselves and other road users, norexperience congestion costs from the private traffic.

Let a function T(r) denote the car traffic intensity at any locationr∈ [0, R], measured by the number of cars passing through thislocation in the direction of the city center. For rN1 this traffic consistsof car (only) users and P+R users, whereas for rb1 only the car usersare accounted for. We define the congestion costs in two steps.

First, we define an instantaneous disutility e(r) of an individualfrom passing location r, as a function of this location and the trafficintensity, i.e.:

e rð Þ = τ r; T rð Þð Þ:

The congestion cost function τ(r,T) is assumed to be non-decreasingin T, and sufficiently smooth in both arguments. The explicitdependence of τ on r makes the congestion externality dependenton the location.

The instantaneous disutility function e(r) can also be viewed as afunction that, for any given value of r, maps any traffic function T(x)into the resulting disutility at location r

e r; Tð Þ = τ r;∫R

0

δ x−rð ÞT xð Þdx !

;

where δ(x) is the Dirac delta-function. We will use this notation torefer explicitly to the underlying traffic intensity function T.3

Second, we define gross disutilities E(C) and E(P+ R) of the car andP+R users correspondingly, which reside at and hence, depart fromlocation r. Their cumulative disutilities due to congestion along theroute are

E P + Rð Þ r; Tð Þ≡∫r

1

e x; Tð Þdx; and

E Cð Þ r; Tð Þ≡∫r

0

e x; Tð Þdx = ∫1

0

e x; Tð Þdx + E P + Rð Þ r; Tð Þ:

The utility of public transport users is given by the same expres-sion (1) as in the basic model. The utilities of the car and P+R usersare as follows:

U Cð Þ = U0;i−ripi−c−E Cð Þ ri; Tð Þ and

U P + Rð Þ = U0;i− ri−1ð Þpi−s−ti−E P + Rð Þ ri; Tð Þ:

Thus, the traffic in an interval has a negative impact on the utilityof individuals that pass through that interval by car. Using thenormalization (2) and defining

c Tð Þ≡ c−að Þ + ∫1

0

e x; Tð Þdx

allows us to write individuals net utilities as follows:

U Pð Þ = 0; U Cð Þ r;u; Tð Þ = ru−c Tð Þ−E P + Rð Þ r; Tð Þ; andU P + Rð Þ r;u; Tð Þ = r−1ð Þu−s−E P + Rð Þ r; Tð Þ:

The congestion externality affects individuals' utility specificationin two ways. In case of congestion, driving in the periphery bringsabout an extra origin-dependent cost of E(P+R)(r,T). Congestioninside the city raises city center driving cost by ∫1

0e x; Tð Þdx, which isorigin-independent.


As in the basic model, we define the indifferent travelers, and itfollows that

u P=Cð Þ r; Tð Þ = c Tð Þ + E P + Rð Þ r; Tð Þr

; u P + R=Cð Þ r; Tð Þ = c Tð Þ−s; and

u P=P + Rð Þ r; Tð Þ = s + E P + Rð Þ r; Tð Þr−1

:

Let us fix r and consider individuals residing at r. It can be seen thatonly individuals with

u∈ u P=P + Rð Þ r; Tð Þ;u P + R=Cð Þ� �=

s + E P + Rð Þ r; Tð Þr−1

; c Tð Þ−s

!

prefer P+R to the car and public transport. Therefore, if

u P=P + Rð Þ r; Tð Þ N u P + R=Cð Þ r; Tð Þ

there are no such individuals. Thus, the demands D Cð Þ r; Tð Þ andD P + Rð Þ r; Tð Þ for the car and P+R from individuals residing at r can bewritten as follows:

DCð Þ r; Tð Þ = ∫

u

max u P + R= Cð Þ r;Tð Þ;u P= Cð Þ r;Tð Þð Þfu u jrð Þdu

= 1−Fu max c Tð Þ−s;c Tð Þ + E P + Rð Þ r; Tð Þ

r

!jr

!; and

D P + Rð Þ r; Tð Þ = ∫u P + R= Cð Þ r;Tð Þ

min u P + R= Cð Þ r;Tð Þ;u P= P + Rð Þ r;Tð Þð Þfu u jrð Þdu

= Fu c Tð Þ−s jrð Þ−Fu min c Tð Þ−s;s + E P + Rð Þ r; Tð Þ

r−1

!jr

!:

Suppose that individuals expect the traffic intensity T rð Þ in

equilibrium. They form demands D Cð Þ r; T� �

and D P + Rð Þ r; T� �

that, in turn, generate the following actual traffic:

T rð Þ = Ω T� �

≡∫R

rD Cð Þ x; T

� �+ D P + Rð Þ x; T

� �� fr xð Þdx; if r N 1

∫R

1

D Cð Þ x; T� �

fr xð Þdx; if r b 1:

8>>>>><>>>>>:

The rational expectationequilibriumof themodel isdefinedas follows.

Definition 1. The traffic intensity function T*(r) is an equilibrium ofthe extended model if and only if it is a fixed point of the mappingΩ(T), i.e., T*=Ω(T*).

In equilibrium, individuals' expectations T rð Þ of the traffic intensitycoincide with the actual traffic intensity induced by their utility-maximizing choices of the mode of transportation. The followingproposition establishes the existence and the uniqueness of equilib-rium, and provides an algorithm for its computation.

Proposition 5. Let the functions fr(r), fu(u|r) be finite, and thefunction e(r,T) be Lipschitz-continuous in T.4 Then, the integralequation T=Ω(T) always has a unique solution T*(r), which can befound iteratively by using the following recursion:

T0 rð Þ≡0; Tk + 1 = Ω Tkð Þ;so that T� rð Þ = limk→∞Tk rð Þ:

4 That is, there exists an MN1 such that for any functions T1, T1: ‖e(r,T1)−e(r,T2)‖≤M‖T1−T2‖, with the supremum norm: ‖T‖=sup r∈ [0,R]|T(r)|.

Proof. The proof is identical to the proof of the Picard–Lindelöftheorem, which establishes the existence and the uniqueness of asolution to the initial value problem in the theory of ordinarydifferential equations. We provide the outline of the proof.

Under the assumptions made, the mapping Ω(T) is Lipschitz-continuous and bounded. Thus, the so-called contraction mappingprinciple, i.e., the Banach fixed point theorem, applies, whichguarantees the existence and the uniqueness of T*(r). The sequence{T

k}, k=0,…∞, defined in the proposition, is the uniformly conver-

gent Cauchy sequence, and its limit is the solution T*(r). □

Although the equilibrium traffic intensity function T*(r) cannot bewritten in a closed form (unless e(r)=const, as in the basic model),Proposition 5 provides an algorithm that allows us to compute T*(r), theresulting modal split, car traffic, and consumer surplus numerically.

This overall impact of P+R is a trade-off between three effects. Inresponse to changes in the costs of P+R, individuals switch from carto P+R and from public transport to P+R, as in the basic model.However, congestion externalities may also yield a third effect on themodal split. In response to changes in the costs of P+R, individualsmay directly switch from public transport to the car, without usingP+R. We illustrate this effect for the case cNsN0.

Let us denote equilibrium traffic intensity T*(r) with and withoutP+R available by T+ and T− correspondingly, and assume that in theabsence of P+R, c(T−)N0 and, hence, sNc(T−)N0. Fig. 6 (left)schematically illustrates the modal split in this case under someauxiliary assumptions.5 Similar to the modal split without thecongestion externality, which is shown in Fig. 4, all individualsabove the curve u(P=C)(r,T−) use the car whereas the individualsbelow it use public transport. Contrary to what Fig. 4 shows, however,the introduction of P+R affects the indifference curve u(P=C)(r,T−)due to its effect on the traffic intensity T*(r).

Let us assume for a moment that after the introduction of P+R, allindividuals keep using the traffic intensity T− for making their modalchoices, as in Fig. 4. One implication is that some former car users,denoted by the vertically shaded area in Fig. 4, switch to P+R andpark their cars at the city border. At the same time, as the indifferencecurve u(P= C)(r,T−) is assumed to remain intact, there are noadditional car users. As a result, the car traffic inside the citydecreases, which leads to a decrease in the cost of driving in the cityc(T+) below c(T−). As the city becomes less congested, the car attractssome public transport users, in particular those who reside close tothe city border. This explains why the indifferent curve u(P=C)(r,T)shifts downwards after the introduction of P+R. Fig. 6 (right)illustrates the resulting modal split.

The P+R users come from either former car users, denoted by thevertically shaded area, or public transport users, denoted by thehorizontally shaded area. Fig. 6 (right) also illustrates the third effect.Due to lowering of the indifference curve u(P=C)(r,T+), there is a setof individuals, shown by the gray area, who switch from publictransport directly to the car.

This effect is similar for the case when c(T−)b0. In this case, theindifference curve u(P=C)(r,T+) also passes below u(P=C)(r,T−) (andfurther away from the origin). However, this effect disappears whenthe cost of using P+R become significantly lower that c(T+). Fig. 3shows why this is the case: all individuals who are indifferentbetween the car and public transport, i.e., individuals along the curveu(P=C)(r,T), are already P+R users. This is another reason why P+Rshould be made as cheap and efficient as possible if it is used as atraffic reducing measure.

The balance of these three effects, that is, switching from car toP+R, from public transport to P+R, and from public transport to car,

5 These are assumptions that guarantee that indifference curves u(P=P+R)(r,T) andu(P=C)(r,T) are downward-sloping and crossing the horizontal indifference curveu(P+R=C)(r,T) at r*bR. The assumption de/dr≤0 suffices for our purposes.

Fig. 6. Modal split before (left) and after (right) introduction of P+R.


determines the net effect of P+R on themodal spit. Depending on theinstantaneous disutility function e(r,T), the net effect can be eitherpositive or negative, even for the uniform distribution. Proposition 5allows for a numerical computation of this net effect, as well as theimpact on welfare. This requires an empirical investigation.

3.7. Endogenous demand for transportation

In the preceding analysis, we focus on individuals who alreadytravel by car or public transport before the introduction of P+R. It isclear, however, that P+R may also induce travel from individualswho did not go into the city before. In this sub-section, we discuss theeffects of P+R on this additional traffic.

In the absence of congestion externalities, new traffic can onlybe fromnewP+Rusers. The cost of using P+R sdoes not affect utilitiesU(P) andU(C) so that all new traffic stops at P+R.Hence, the basicmodelpredicts traffic in the city correctly, even with endogenous demand.

Endogenous demand increases traffic in the periphery. Thisrequires an empirical or computational model, as in the case ofcongestion externalities. Marginal changes in s only affect individualswho (i) do not travel by car and public transport, i.e., U(P)b0, U(C)b0,and (ii) indifferent between staying home and using P+R, i.e.,

U P + Rð Þ = U0;i− ri−1ð Þpi−s−a −ti = 0:

Hence, this effect depends on the distribution of the term U0, i−(ri−1)pi− ti, which contains four generic individuals' character-istics. This multiplicity does not allow us to generalize the modelanalytically and requires an empirical or computational approach.Another reason that motivates an empirical rather than an analyticalinvestigation is that even when the effect of P+R on the additionaltraffic is computed, its impact on social welfare is not straightfor-ward, as one has to trade off the costs of additional traffic againstthe economic benefits of having more visitors in the city. This is anempirical question.

4. Conclusion

We have analyzed P+R at the edge of cities and towns that aim tointercept motorists from traveling into the city. In general, such P+Rfacilities may serve two rather different but not mutually exclusivepurposes. First, P+R allows individuals to avoid driving their car intothe city and to save on costs caused by congestion and parking. Thisreason makes P+R only attractive for individuals with a preferencefor private car. Second, P+R allows individuals to avoid slow andlow-frequency local public transport services at their location ofresidence, and use P+R to access directly the mainline public

transport network. This reason makes P+R only attractive forindividuals with a preference for public transport. P+R may alsoserve both purposes at the same time.

In the absence of congestion externalities, the effect of P+R ontotal private car traffic is fully determined by the distribution ofindividuals' preferences over car and public transport. In all instances,opening a P+R facility reduces total car traffic if the distribution ofindividuals' net utilities is weakly increasing in utility. This traffic-reducing effect of P+R is weaker if P+R only provides a cheaperalternative to city driving than if P+R also provides cheaper access tothe public transport network. In the presence of the first motive,opening a P+R facility decreases car traffic under any increasingdensity function, but in the presence of the second motive, the trafficeven decreases if the corresponding distribution density function“slightly” decreases. The positive effect of P+R on social welfare goesbeyond this reduction in traffic due to its positive effect on consumersurplus.

Assuming a unimodal, e.g., normal, distribution of individuals' netutilities (opportunity costs of public transport) and assuming that themajority of individuals already travel by car before opening a P+Rfacility, immediately implies that the density in the relevant range isincreasing, and, consequently, that P+R reduces car traffic. Althoughthese assumptions seem plausible on average, they may not apply toall situations. Hence, an empirical investigation of the distribution ofindividuals' preferences is required in order to determine the impactof opening a specific P+R facility.

Acknowledgement

We thank Yves Zenou and two anonymous referees for helpfulcomments and suggestions.

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park-and-ride: good for the city, good for the region?

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